We briefly give some comments on our proof. In [1]}, the authors make use of a families of Beltrami flows to control the interference terms between different waves in the construction. For the Boussinesq system, we do not know if there exist the analogous special solutions. Following an idea in [2]}, [3]}, we make use of a multi-steps iteration scheme and one-dimensional oscillation. More precisely, in each step, we add some plane waves which oscillate along the same direction with different frequency, and thus only remove one component of stress error in each step. To reduce the whole stress errors, we divide the process into several steps. On the other hand, since the velocity and temperature are coupled together in Boussinesq system, we need to reduce two stress errors simultaneously. To achieve this, we need to extend the geometric lemma in [4]} and add associated plane waves in the velocity and temperature simultaneously in each step. Their coordination is important for us to reduce the temperature stress error and construct continuous temperature.
